Editing SSC/Stability/UniformDensity (section)

===Sterne's Presentation===
In what follows, as before, text presented in a green font has been taken verbatim from {{ Sterne37hereafter }}.  He begins by writing the unknown eigenfunction as a power series expanded about the origin, specifically,
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<math>\xi_1</math>
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<math>=</math>
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<math>\sum\limits_{0}^{\infty} a_k x^k \, ,</math>
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with, <math>a_0 = 1</math>.  <font color="green">It is found by substitution that the terms in odd powers of <math>x</math> vanish, and that the coefficients of the even terms satisfy the recurrence formula</font>,

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<math>a_{k+2}</math>
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<math>=</math>
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<math>a_k \cdot \frac{k^2 + 5k - \mathfrak{F}}{(k+2)(k+5)} \, .</math>
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The wave equation and attending boundary conditions will all <font color="green">be satisfied if we choose <math>\mathfrak{F}</math> so as to make the series solution terminate with some term, say the <math>2 j^\mathrm{th}</math> where <math>j</math> is zero or any positive integer.  This it will do</font> [via the above recurrence relation] if,
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<math>\mathfrak{F} = 2j(2j+5) \, .</math>
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The first few solutions are displayed in the following boxed-in image that has been extracted directly from &sect;2 (p. 587) of {{ Sterne37 }}; to the right of his table, we have added a column that expressly records the value of the square of the normalized eigenfrequency that corresponds to each of the solutions presented by {{ Sterne37hereafter }}.

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Table of exact eigenvector expressions extracted from &sect;2 (p. 587) of &hellip;<br />
{{ Sterne37figure }}
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<math>\frac{n^2}{4\pi G \bar\rho}</math>
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<!-- [[File:Sterne1937SolutionTable1.png|600px|center|Sterne (1937)]] -->
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    <td align="right"><math>j=0 \, ;</math>&nbsp; &nbsp; &nbsp;</td>
    <td align="right"><math>\mathfrak{F}=0 \, ;</math>&nbsp; &nbsp; &nbsp;</td>
    <td align="right">&nbsp;<math>\xi_1 = 1</math></td>
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  <td align="center"><math>\gamma - 4/3</math></td>
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    <td align="right"><math>j=1 \, ;</math>&nbsp; &nbsp; &nbsp;</td>
    <td align="right"><math>\mathfrak{F}= 14 \, ;</math>&nbsp; &nbsp; &nbsp;</td>
    <td align="right"><math>\xi_1 = 1 - (7/5)x^2</math></td>
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  <td align="center"><math>2(5\gamma - 2)/3</math></td>
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    <td align="right"><math>j=2 \, ;</math>&nbsp; &nbsp; &nbsp;</td>
    <td align="right"><math>\mathfrak{F}= 36 \, ;</math>&nbsp; &nbsp; &nbsp;</td>
    <td align="right"><math>\xi_1 = 1 - (18/5)x^2 + (99/35)x^4</math></td>
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  <td align="center"><math>7\gamma - 4/3</math></td>
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    <td align="right"><math>j=3 \, ;</math>&nbsp; &nbsp; &nbsp;</td>
    <td align="right"><math>\mathfrak{F}=66 \, ;</math>&nbsp; &nbsp; &nbsp;</td>
    <td align="right"><math>\xi_1 = 1 - (33/5)x^2 + (429/35)x^4 - (143/21)x^6</math></td>
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  <td align="center"><math>12\gamma - 4/3</math></td>
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